Local Space does not work with Transformations Modifier
(Vocabulary correction: it's a constraint, not a modifier).

Your constraint setup looks fine, it should work as you want.
I recommend you check again the local axes. Especially the bone, since the local axes of a bone do not correspond to the scene like for every other type of object.

First, set your Transform orientation to Local :



Then select your object in object mode, it should show you right away.

For the bone, it's a little bit different. To check a bone's local axes, you have to switch to pose or edit mode first, then select it. (Or enable the axes display option of the armature.)